The complexation between a polyion and an oppositely charged spherical macroion in the framework of the primitive model has been studied by the use of Monte Carlo simulations. The polyion length, linear charge density, and bare persistence length are varied systematically, while the properties of the macroion are kept constant. The polyion charge to macroion charge ratio is varied between 1/4 and 4. The structure of the complex is investigated by direct visualization; polyion bead complexation probability; loop, tail, and train characteristics; degree of overcharging; and tail joint probability functions. The strongest complexes are observed for flexible chains, where the polyion is folded around the macroion. In the case of fully flexible chains, a transition from a collapsed state to a fluctuating two-tail state and eventually to a one-tail state are observed as the chain length is increased. As the stiffness is increased, several complex structures, such as multiloop, single-loop, and solenoid arrangements, and finally a structure involving only a single contact between the polyion and the macroion occur. In particular, for long and highly charged polyions, a transition from the one-tail state to a two-tail state appears as the chain stiffness is increased. A discussion with recent theories and other simulation studies is also provided.